Open Access
2016 First order deformations of pairs and non-existence of rational curves
Bin Wang
Rocky Mountain J. Math. 46(2): 663-678 (2016). DOI: 10.1216/RMJ-2016-46-2-663


Let $X_0$ be a smooth hypersurface (assumed not to be generic) in projective space $\mathbf {P}^n$, $n\geq 4$, over complex numbers, and $C_0$ a smooth rational curve on $X_0$. We are interested in deformations of the pair $C_0$ and $X_0$. In this paper, we prove that, if the first order deformations of the pair exist along each deformation of the hypersurface $X_0$, then $\deg (C_0)$ cannot be in the range \[ \bigg ( m\frac {2\deg (X_0)+1}{\deg (X_0)+1}, \frac {2+m(n-2)}{2n-\deg (X_0)-1}\bigg ), \] where $m$ is any non negative integer less than \[ \dim (H^0(\mathcal {O}_{\mathbf {P}^n}(1))|_{C_0} )-1. \]


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Bin Wang. "First order deformations of pairs and non-existence of rational curves." Rocky Mountain J. Math. 46 (2) 663 - 678, 2016.


Published: 2016
First available in Project Euclid: 26 July 2016

zbMATH: 1356.14032
MathSciNet: MR3529086
Digital Object Identifier: 10.1216/RMJ-2016-46-2-663

Primary: 14J70 , 14N10 , 14N25

Keywords: Hypersurface , normal bundle , Rational curve

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 2 • 2016
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